“What I can say, as an objective fact, is that there has been no evidence produced to substantiate a claim like that,” he told reporters at a White House briefing.

Mr Trump, who won the all-important electoral college count, aired his grievances with the election result in a tweet on Sunday.

“In addition to winning the Electoral College in a landslide, I won the popular vote if you deduct the millions of people who voted illegally,” he wrote.

‘For Trump, controversy is like water to a fish’

Analysis: Anthony Zurcher, BBC North America reporter

It’s a classic Donald Trump move. Take an accusation, and turn it on an accuser. His opponents want a recount in states he won? Then he’ll allege massive voter fraud in states carried by Hillary Clinton.

There is, of course, no evidence of the “millions” of illegal votes that Mr Trump says were cast for Democrats. If there were, it would merit a full investigation and not a series of Sunday-morning tweets from the president-elect. The veracity of these accusations seems of little import to Mr Trump.

What matters is that by going on the offensive, he turns a story about the legitimacy of his narrow wins in key states into a muddled mess. As he has done in the past, he raises the volume in hopes of drowning out a negative story.

The irony is that, in this case, it seems a pointless undertaking. The Green Party-funded recounts will almost certainly fail to reveal electoral malfeasance. Mr Trump could have let them proceed without comment and avoided any controversy.

Then again, for this president-elect, controversy is like water to a fish. It surrounds and sustains him. Perhaps he can’t function without it.

The president-elect’s Twitter outburst comes after the Clinton camp said it would support a vote recount in Wisconsin initiated by Green Party candidate Jill Stein.

Ms Stein also notified the elections board in Michigan, where Mr Trump’s 16 electoral votes were certified on Monday, that it would seek a statewide recount of the presidential election results.

Her campaign moved to do the same in Pennsylvania.

Mr Trump won by two-tenths of a percentage point out of nearly 4.8 million votes, making it the closest presidential race in Michigan in more than 75 years.

He is the first Republican presidential nominee to win Michigan since 1988.

Ms Stein’s recount effort was driven by the #recount2016 social media campaign, which has raised over $6.3m (£5m).

But, he added, “we have an obligation to the more than 64 million Americans who cast ballots for Hillary Clinton to participate in ongoing proceedings to ensure that an accurate vote count will be reported”.

Also on Monday, Mr Trump met with former CIA Director and retired Army General David Petraeus, who was convicted of sharing classified information with his lover in 2012.

Mr Petraeus is reportedly being considered for the role of Secretary of State, the nation’s top diplomat.

Mrs Clinton was criticised by Mr Trump during the campaign for her handling of classified information during her time as Secretary of State.

Despite calling her handling of the classified information “extremely careless”, the FBI never found Mrs Clinton’s behaviour to be criminal.

Here’s how some of the main news outlets reacted to Mr Trump’s claims on Twitter:

New York Times: “President-elect Donald J. Trump said on Sunday that he had fallen short in the popular vote in the general election only because millions of people had voted illegally, leveling the baseless claim as part of a daylong storm of Twitter posts voicing anger about a three-state recount push.”

Washington Post: Mr Trump was accused of “parroting a widely debunked conspiracy theory” and “suggesting without evidence that millions of people illegally”. The newspaper also ran a fact-checking article which concluded: “This is a bogus claim with no documented proof.”

Fox News: The conservative broadcaster did not draw attention to the baselessness of Mr Trump’s claim until halfway through the fifth paragraph, when it conceded that “the magnate offered no proof of the alleged irregularities”.

Huffington Post: The online outlet criticised media for reporting his allegations without challenging them: “Too often, news organizations amplify Trump’s assertions in headlines with some variation of “Trump tweets” or Trump claims” or “Trump says” ― whether or not those assertions are true.”